Solution Of Equation Research Articles

THREE-COMPONENT VERSION OF THE TIKHONOV REGULARIZATION METHOD FOR OPERATOR EQUATIONS OF THE FIRST KIND

t An ill-posed problem in the form of a linear operator equation is considered. It is assumed that the solution to the equation in the one-dimensional case can be represented in the form of a sum of three components: the first component contains discontinuities, the second contains discontinuities in the derivative, and the third is continuous. To construct a stable approximate solution, the three-component Tikhonov method is used. In this case, the stabilizer is the sum of three functionals: BVp-norm of the first component, BVp-norm of the derivative for the second component and the norm of the Sobolev space for the third component, and each functional depends on only one component. The convergence of the sum of regularized components to the solution of the original equation is proved. In addition, piecewise uniform convergence of approximate solutions is established. The results of numerical experiments on reconstructing a three-component model solution for the Fredholm equation of the first kind are presented.

ON THE PROPERTIES OF A FAST ALGORITHM FOR SOLVING A THREE-DIMENSIONAL INVERSE PROBLEM OF SCALAR ACOUSTICS

t Theinverse single-frequency problem of scalar acoustics in three-dimensional space is considered. It consists in determining the characteristics of acoustic inhomogeneities lying in a flat layer from the distribution of the complex amplitude of the acoustic field in a flat recording layer. To effectively solve it, a special numerical algorithm is used, previously proposed and justified by the authors. As its component part, the algorithm uses solutions to special one-dimensional Fredholm equations of the 1st kind, and in the case of ambiguity in the solution of the latter, their normal solutions (solutions with a minimum norm) are sought. To regularize this ill-posed inverse problem, Tikhonov regularization and the TSVD method are used. A systematic numerical study is carried out of the influence of various parameters in the data recording scheme on the accuracy of the approximate solutions obtained using the algorithm. In particular, we study the dependence of this accuracy on the position of point sources and composite sources that cause sound vibrations, its dependence on the position of the data recording layer, and on the number of planes in which the recording sensors lie. It is shown that acceptable accuracy of approximate solutions can be obtained even with two such layers. In addition, an approach to the numerical study of the ambiguity of the solution to the inverse problem under consideration is proposed. It is based on special theoretical statements given in the article, and on the same numerical algorithm for solving the inverse problem. The approach is focused on comparing several normal solutions of one-dimensional integral equations from the problem under consideration. These normal solutions are calculated with respect to various elements. An example of using this theory to numerically estimate the non-uniqueness of a solution is given.

Simplicial epidemic model with individual resource

The investment and rational use of medical resources are crucial for the prevention and control of epidemics. Here we propose a coupled dynamics model of individual resource and disease transmission on simplicial complexes that considers the dynamic changes of resource quantity during the spreading of the epidemic. Firstly, we derive the solutions (infection density) of the disease transmission dynamics equations under the mean-field approximation, and analyze their stability. We find that the number of stable solutions is related to the infection rates of both interactions and higher-order interactions, and we obtain the transmission threshold under different initial infection densities. Secondly, in the numerical simulation, we discover a double hysteresis loop of infection density as it evolves with the infection rate of interactions. Moreover, we find a critical value for the cost of disease recovery; when the recovery cost is above this critical value, the system state transitions from partial infection to complete infection. Interestingly, we find a maximum value for infection density when the system is in partial infection state. Finally, we verify the existence of the double hysteresis loop on real-world networks.

THREE-COMPONENT VERSION OF THE TIKHONOV REGULARIZATION METHOD FOR OPERATOR EQUATIONS OF THE FIRST KIND

t An ill-posed problem in the form of a linear operator equation is considered. It is assumed that the solution to the equation in the one-dimensional case can be represented in the form of a sum of three components: the first component contains discontinuities, the second contains discontinuities in the derivative, and the third is continuous. To construct a stable approximate solution, the three-component Tikhonov method is used. In this case, the stabilizer is the sum of three functionals: BVp-norm of the first component, BVp-norm of the derivative for the second component and the norm of the Sobolev space for the third component, and each functional depends on only one component. The convergence of the sum of regularized components to the solution of the original equation is proved. In addition, piecewise uniform convergence of approximate solutions is established. The results of numerical experiments on reconstructing a three-component model solution for the Fredholm equation of the first kind are presented.

Existence of Solutions for Impulsive Neutral Semilinear Evolution Equations with Nonlocal Conditions

Existence of Solutions for Impulsive Neutral Semilinear Evolution Equations with Nonlocal Conditions

Central cone film cooling scheme of turbofan engine based on multi-fidelity simulation

To improve the computational efficiency of multi-fidelity simulation, an improved fully coupled method was proposed, which was applied to the coupled simulation of a 3-D model of an exhaust system and a 0-D model of a turbofan engine. Different from conventional approach, by “truncating” the engine model, the CFD calculation of the exhaust system can be decoupled from the solution of nonlinear equations in the engine model, and an outer iteration equation set was constructed to solve the multi-fidelity model. It was found that the improved fully coupled method has better computational efficiency than traditional methods while maintaining equivalent computational results. Then, the improved fully coupled method was applied to the design of the central cone film cooling system, the influence of cooling gas bleed mechanism, the hole open area ratio of the central cone and hole density, hole direction, hole shape on the infrared characteristics was investigated. The impact of central cone film cooling on the overall performance of the engine was quantitatively evaluated. Combined with various favorable factors, when maintaining the HP rotor speed unchanged, the integrated infrared radiation intensity of the central cone under supersonic cruise conditions is reduced by 93% compared to the uncooled state, while the engine thrust increases by 0.3% and fuel consumption rate increases by 0.18%. The multi-fidelity simulation method proposed in this paper and the conclusions obtained are of great significance for the structural design of low-infrared exhaust systems.

Use of GeoGebra as a didactic resource for the solution of linear ordinary differential equations

Use of GeoGebra as a didactic resource for the solution of linear ordinary differential equations

On Meromorphic Solutions of Some Fermat-Type Functional Equations

On Meromorphic Solutions of Some Fermat-Type Functional Equations

The Numerical Solutions of 3-Dimensional Fractional Differential Equations

The Numerical Solutions of 3-Dimensional Fractional Differential Equations

Analytical investigation of vesicle dynamics via the modified Riemann-Liouville fractional derivative: Mittag-Leffler function solution and comparative analysis with Caputo's derivative.

The solution of fractional differential equations is a significant focus of current research, given their prevalence in various fields of application. This paper introduces an innovative exploration of vesicle dynamics using Jumarie's modified Riemann-Liouville fractional derivative within a five-dimensional fractional rigid sphere model. The study reveals an exact solution through the Mittag-Leffler function, providing a deep understanding of intricate vesicle dynamics, including alternative motions, such as tank-treading with over-damped and under-damped vesicle oscillations, respectively, TT-OD and TT-UD. A comparative analysis with Caputo's derivative emphasizes the effectiveness of these fractional derivatives, contributing not only to theoretical insights but also practical implications in applied mathematics and biophysical systems. The findings advance our understanding of complex vesicle behaviors, particularly in mimicking real cell-like behaviors, and pave the way for further research and applications in the field.

Observational constraints on the wet dark fluid model in the fractal gravity

A flat Friedmann–Robertson–Walker (FRW) cosmological model with Wet Dark Fluid has been studied based on fractal gravity. The exact solution of the field equations is obtained using linear time-varying deceleration parameter (Akars̈u and Dereli, 2012) and assuming fractal parameter ξ≠0. The model parameters involved in the model have been constrained using the Hubble datasets H(z) of 77 data points, known as cosmic chronometers (CC), recently published Pantheon, and the joint data (CC+Pantheon samples). Additionally, we compare our model with ΛCDM in standard cosmology via error bar trajectories. We study the physical and the cosmographic parameters, such as the Hubble parameter H(z), deceleration parameter q(z), and the higher derivatives of the deceleration parameter q(z), etc. under cosmic observations. It is observed that our model transits from a decelerating state to an accelerating state at ztr≈2.764. The other astrophysical parameters such as the jerk parameter and snap parameter are also discussed. Finally, we conclude that our model is an accelerating quintessence dark energy model.

Multiplicity of solutions for anisotropic Dirichlet problem with variable exponent

We establish some results on the existence of multiple nontrivial solutions for a general anisotropic elliptic equations. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces,combined with adequate variational methods and a variant of the Mountain Pass lemma.

Approximate solution of time-fractional non-linear parabolic equations arising in Mathematical Physics

Approximate solution of time-fractional non-linear parabolic equations arising in Mathematical Physics

Time-fractional fabric to quantify non-Fickian diffusion in porous media: New vision from previous studies

The diffusion kinetics of different substances in porous media is a priori described by the second Fick's law of diffusion. Herein, we demonstrate that such a description sometimes yields erratic results. In contrast, the time-fractional diffusion equation is found to reflect the diffusion kinetics in the case of Fick's law failure. The analytic solutions of the time-fractional diffusion equation are based on the Mittag–Leffler function. At large times, it is shown that the utilization of the approximation of the Mittag–Leffler function for the large values of its argument for the description of the experimental data gives almost the same results as those obtained using the Mittag–Leffler function itself. The time-fractional diffusion is applied on a phenomenological basis. Nevertheless, we speculate that the deviations from Fick's law may be governed by strong adsorption of the diffusing species inside the small micropores of a porous material.

The ultimate boundedness of solutions for stochastic differential equations driven by time-changed Lévy noises

The asymptotical ultimate boundedness is established for SDEs driven by time-changed Lévy noises not just in the pth moment sense but in the almost sure sense. In particular, we would like to point out that the time-changed process is the inverse of general Lévy subordinators.

Nonlinear dynamic wave characteristics of optical soliton solutions in ion-acoustic wave

Traveling wave solutions are utilized to depict reaction-diffusion and analyze electrical signal transmission and propagation in space-time nonlinear fractional order partial differential equations like the space-time fractional Telegraph and Kolmogorov-Petrovsky Piskunov equations, and that are used in physical science to model combustion, biological research to model nerve impulse propagation, chemical dynamics to model concentration in order wave propagation, and plasma to model the progression of a set of duffing oscillators. In this study, the new generalized (G′/G)−expansion technique was employed to construct some novel and more universal closed-form traveling wave solutions in the sense of conformable derivatives which explain the above-stated phenomena properly. By utilizing complex fractional transformation, the ordinary differential equations are generated from fractional order differential equations. The recommended technique allowed us to produce some dynamical wave patterns of kink, single soliton, compacton, periodic shape, multiple periodic waves, anti-kink, and other structures are developed, which are shown using 3D plots and contour plots to illustrate the physical layout clearly. The traveling waveform responses can be defined in terms of functions based on trigonometry, hyperbolic operations, and rational functions and that are quick, flexible, and simple to reproduce. Furthermore, the obtained closed-form solutions for nonlinear fractional evolution equations make stability analysis and accuracy comparison amongst numerical solvers easier, which asserts that the new generalized (G′/G)−expansion technique is one of the most proficient and effective approaches.

Unified model modal expansion method for full field reconstruction and inversion under mechanical and thermal loading

In this paper, a novel method termed UMM-T&S (Unified Model Modal reconstruction and inversion method for Temperature field and Structural field under limited sensors) is proposed. This method aims to enhance high-precision structural performance inversion and reconstruction under mechanical and thermal loading conditions, particularly in the context of structural health monitoring within the aviation and aerospace sectors. The approach begins by deriving the “temperature modal” solution of the heat conduction equation, drawing upon the modal expansion techniques used in multi-degree-of-freedom systems. This is followed by constructing a parameter mapping relationship within the unified model, enabling the solution of both the structural field (encompassing strain and displacement) and the temperature field through a single structural model modal expansion. Furthermore, the paper details a numerical validation using two different temperature fields and a thermal loading experimental verification on a stiffened plate under five distinct working conditions. The proposed UMM-T&S method yields lower errors (ranging between 10%-40%) and more stable performance across various complex working conditions, in comparison to existing methods. The analysis also delves into the sources of errors and their cumulative impacts.In conclusion, the innovative UMM-T&S method offers a swift, precise, and robust solution for the intricate task of reconstructing and inverting structural performance under combined mechanical and thermal stresses. The method's efficacy and potential are substantiated through experimental validation and comparative analysis against an established methodology, underscoring its capability to effectively tackle the challenges faced in structural monitoring under such demanding conditions.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra–Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schläfli integral of the Volterra–Bessel function. In this respect, we study the asymptotic analysis of the Volterra–Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

Integration of the Korteweg-de Vries equation with time-dependent coefficients in the case of moving eigenvalues of the Sturm–Liouville operator

The inverse scattering method is used to integrate the Korteweg-de Vries equation with time-dependent coefficients. We derive the evolution of the scattering data of the Sturm–Liouville operator whose coefficient is a solution of the Korteweg-de Vries equation with time-dependent coefficients. An algorithm for constructing exact solutions of the Korteweg-de Vries equation with time-dependent coefficients is also proposed; we reduce it to the inverse problem of scattering theory for the Sturm–Liouville operator. Examples illustrating the stated algorithm are given.